The following question concerns Deligne-Mumford compactification of the (coarse) moduli space $M_{g,n}$ of smooth complex genus g curves with $n$ marked points.

If there are no marked points (ie $n=0$), then given a family of curves $C_\lambda$ given by $w^N=z^a (z-1)^b (z-\lambda)^c,$ as $\lambda\to 0$ I believe the limit is the stable curve $w^N=z^{a+c} (z-1)^b.$

However, if all lifts of $0, 1, \lambda, \infty$ are marked points, I no longer know how to determine the limit as $\lambda\to0$ in $\bar{M_{g,n}}$.

So, my question is, how do you do such explicit computations when there are marked points?